SECTION 1
DATE	November 23 (Monday)
TIME	8:00-9:05
PLACE	SCI. 3610

Examination Type: Closed notes and books. But you will be allowed to use one sheet of paper (information sheet) with the formulas and facts that you need (This sheet should not have solutions of problems or examples)

Coverage: Section 5.1 - 7.1 (included)

5. From Probability to Inference

5.1 Counts and Proportions

How to use binomial tables to find probabilities

Mean and variance of counts and sample proportions

Finding count and sample proportion probabilities by using normal approximation

5.2 Sample Means

Determining sampling distribution of sample mean for normal and nonnormal populations

The mean and standard deviation of the sample mean

Use of central limit theorem

6. Introduction to Inference

6.1 Estimating with Confidence

Constructing a level C confidence interval for m

Determination of sample size for a given margin of error m

6.2 Tests of Significance

Setting up null and alternative hypotheses

P-value

Tests with fixed significance level

6.3 Use and Abuse of Tests (excluding Power and Inference as Decision)

Power (pg. 477) not included

Inference as decision (pg. 481) not included

7. Inference for Distributions

7.1 Inference for the Mean of a Population

t distribution

confidence interval for m when s is unknown (assumptions)

Testing a hypothesis for m when s is unknown (assumptions) (P-value or fixed significance level)

The power of t test (pg. 511) not included

Inference for nonnormal populations (pg. 512) not included

1. A particular type of birth-control pill is 90% effective. A random sample of 20 persons are selected. a. What is the probability distribution of x, the number of unplanned births for this sample? Justify your answer? b. How many unplanned births would you expect in this sample? What is the standard deviation of the random variable defined in part (a)? c. Find the probability that exactly 3 out of 20 will have unplanned births? d. Use the normal approximation to find the probability that in the sample the the pill will be effective more than 80% of the time.

2. 40% of all fire alarms in a certain neighborhood are false alarms. A random sample of 12 fire alarms is selected. a. What is the mean number of false alarms in the sample? What is the standard deviation of this number? b. What is the probability that exactly four of them are false alarms? c. What is the probability that at least four of them are false alarms? d. What is the probability that at most two of them are false alarms? e. What is the probability that none of them are false alarms? f. Use the normal approximation to find the probability that more than 60% of the fire alarms in the sample are false alarms.

3. The speeds of all cars traveling on a stretch of Interstate Highway I-95 are normally distributed with a mean of 68 miles per hour and a standard deviation of 3 miles. a. What is the probability that a single traveler randomly chosen from all travelers is violating the 65 miles speed limit? b. If a random sample of 36 cars traveling on this highway has been selected, what is the probability that their average speed will exceed the 65 miles speed limit?

4. In 1987, the average annual rate of interest paid by savings and loan institutions in Pennsylvania was 7.26%. Assume a normal distribution and a standard deviation of 1.50% to answer the following questions. a. What is the probability that a randomly selected Pennsylvania savings and loan institution paid between 7.00% and 8.00% interest on deposits? b. What is the probability that a randomly selected 9 such institutions on the average paid between 7.00% and 8.00% interest on deposits?

5. The number of violent crimes per day in a certain city possesses a mean equal to 1.3 and a standard deviation equal to 1.7. A random sample of 50 days is observed, and the daily mean number of crimes for this sample, is calculated. a. What is the approximate distribution of sample mean according to the central limit theorem? (Specify its mean and standard deviation.) b. What is the approximate probability that sample mean is less than 1? c. What is the approximate probability that sample mean is more than 1.9?

6. A state game protector collects measurements on the weights of species of fish in a lake. The standard deviation of the weights is known to be 2.13 pounds. A total of 49 measurements are obtained and it is determined that sample mean is 7.34. a. Give a 95% confidence interval for the average weight of that species. b. How many measurements must be averaged to get a margin of error of 0.5 with 95% confidence?

7. The manufacturer of a certain foreign car sold in the United States claims that it will average 35 miles per gallon of gasoline with the standard deviation of 9.3 miles. To test this claim, a consumer's group randomly selects 40 of these cars and drives them under normal driving conditions. These cars average 28 miles to the gallon. Does this sample indicate that average mileage is more than 35? a. State the null and alternative hypotheses? b. Carry out the test and give the P-value. Report your conclusion. c. Is the result significant at the 5% level?

8. Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to estimate its average number of unoccupied seats. To accomplish this, the records of 225 flights are randomly selected, and sample mean is found to be 11.6 seats. Assume that the population standard deviation is known to be 4.1 seats. a. Estimate m= the mean number of unoccupied seats per flight during the past year, using a 90% confidence interval. b. How large a sample needs to be taken to estimate the mean number of unoccupied seats to within 0.35 seats with 90% confidence? c. Test the hypothesis H0: m=13 seats against the hypothesis Ha: m=13 at a=0.01 level. d. Find the P-value of the test.

9. A fast food restaurant is considering a new product. It will be worthwhile to introduce this product if the mean sales per store are more than $600 per week. In a marketing test the product is sold at 26 stores. From the sample it is found that the mean for weekly sales is $603.20 and the standard deviation is $60. a. State the null and alternative hypotheses? b. Carry out the test and give the P-value. Report your conclusion. c. Is the result significant at the 5% level? d. Give a 95% confidence interval for the mean sales.

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